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]]>Some bad ideas that come to mind for the last bit are having multiple parts for the last one, and giving like 20% credit for trying for an hour.

Also it sounds maybe not useless (though probably too confusing) to have multiple linear constraints. Say you want me to do some problems of two different types but the boundaries are fuzzy so several are in between. (Of course the point values would provide info then.)

]]>(Sorry I forgot about WordPress math in the previous comment, and I can’t seem to edit.)

]]>Key Observation: if $A_1,A_2,\dots,A_n$ go counterclockwise around the circle, then the product $\prod_i B_{i,i+1} A_i^{-1}$ is $\exp(i\pi) = -1$, because the sum of the corresponding arcs is precisely half the circumference.

So, if $a_1$ is an arbitrary square root of $A_1$, then there is a **unique** square root $a_2$ of $A_2$ such that $B_{1,2} = -a_1a_2$: namely, $a_2 = -B_{1,2} a_1^{-1}$ works! Continuing in this manner uniquely specifies all $a_i$. The catch is, the choice is consistent if and only if $n$ is odd: we require $(-1)^n$ to be consistent with $-1$ from the Key Observation.

The case $n$ is even (e.g. quadrilateral as you discuss) is similar, except we can no longer take all the signs in the $B_{i,i+1} = \pm a_ia_{i+1}$ to be $-1$. An odd number of the signs must be $+1$ instead.

]]>Now the second part is that I moved to the US. I have learned that there are a lot of competitions held my top universities such as hmmt or pumac. I started to practice them. However, the official rule is that students are not allowed to use them. I have lost my best weapon in this competition but I have a strong desire to do well on them. Is there any advice that you could give to the students who are in similar situation as mine?

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